Math: Postscript

From: David H Kirshner (dkirsh@lsu.edu)
Date: Wed Oct 31 2001 - 11:11:59 PST


Hi XMCA.

I'm REPLYing to Julian Williams' note, but actually responding to the whole
discussion of math, which seems to have run its course, but I didn't have a
chance to respond earlier. ...thanks to those who participated in this
rich strand.

Several aspects of the discussion interest me.

First, in my teaching of mathematics teachers I've found it necessary to
develop a course dedicated exclusively to mathematical enculturation.
That's because many secondary school mathematics teaching candidates,
including many of those who have completed an undergraduate major in
mathematics, do not have a real sense of the mathematical enterprise: what
motivates mathematical exploration; what counts as an interesting problem;
how exploration and proof are dialectically related; how to approach
significant problems; what makes a solution "elegant"; the attraction of
paradox and self-reference, etc. What we do in the course is dabble some in
the philosophy of mathematics (e.g., Lakatos) and interview mathematicians.
But mostly we engage in individual and collective non-routine-problem
solving and problem solving analyses, journaling about our individual
compentencies and dispositions relative to what we see as the cultural
norms of the mathematics community.

An interesting challenge of the process is to come to terms with the
disjunction between what mathematicians do and how they publicly and
privately construct their own work. Going back to a previous posting of
Lemke's, yes, "Mathematics is relations all the way down:"

   [Mathematical] objects? we really mean forms? or relations, no? what a
   mathematician calls an "object" has nothing in common with anything else
   we call an object, does it? Mathematics is relations all the way down,
   no? the "objects" are just placeholders, "arguments" to define the
   qualities of the relations.

But it's also (conceptual) objects all the way up--that's Sfard's theory of
reification: a layering of objects and processes. Mathematicians, it turns
out, locate their own consciousness at the object level, revelling in the
fixedness of their constructed universe (Sfard, 1994). In David and Hersh's
(1981) memorable observation, "The typical working mathematician is a
Platonist on weekdays and a formalist on Sundays" (p. 321). There's
outright hostility in the mathematics community toward
constructivist/psychological discourses about mathematical objects that
don't start and end with the canons of mathematical theory (which, of
course, only encapsulate the object/process relations). So, there are
limits to how much mathematical enculturation one wants to encourage in
future mathematics teachers.

A second interest in the discussion is more specific to Julian's (along
with most other mathematics educators') interest in promoting "maths
teaching based on 'reality', 'applications' and 'modelling'." My concern,
here, is in the presumed incompatibility of formal (rule based)
mathematical manipulations and meaning. Certainly Sfard's experiences of
the problems of promoting reification in schools have made her extremely
cautious about premature formalism:

   While dealing with symbols, the formalists focus on combinations of
   operations (Gregory, 1840). The operations that the high-school student
   is supposed to master, namely those that can be interpreted as a
   generalization of arithmetic calculations, are for the formalists but a
   point of departure, mere inputs to the processes they are really
   interested to investigate. In other words, the formalist's algebra
   begins where the school algebra ends. Besides, although both the
   mathematician and the pupil view the formal operations as aritrary, for
   the formalist such an approach is a matter of a deliberate choice, while
   for the student it is an inevitable outcome of his or her basic
   inability to link algebraic rules on the laws of arithmetic. (Sfard &
   Linchevski, 1994, p. 119)

This seems to imply no legitimate role for formal algebraic approaches to
symbol manipulation until the basic reifications of variable and function
are completed. I struggle with this conclusion which denies important
aspects of rigorous mathematical method to high school students. Rather, I
see the salience of various forms of mathematical activity, and their
relations to one another, as socially mediated, and hence remediable.

I assume these interests are too specialized for XMCA debate, but I wanted
to put them out there.

David Kirshner

dkirsh@lsu.edu
Department of Curriculum & Instruction
Louisiana State University
Baton Rouge LA 70803-4728

"Julian S Williams" <mewssjsw@fs1.ed.man.ac.uk> on 10/16/2001 05:35:25 AM

Please respond to mewssjsw@man.ac.uk

To: xmca@weber.ucsd.edu
cc: (bcc: David H Kirshner/dkirsh/LSU)

Subject: Re: anxiety about the 'ontologic math anxiety'

Im feeling very uncomfortable with the way this conversation has been
going.

First, let me say I spent a lot of time developing maths teaching
based on 'reality', 'applications' and 'modelling'. So I am
sympathetic with much of what martin and bb and others are saying.
And John Holt's thinking about this appealed to me, too.

Yet, the notion of 'reality' needs some thought: whose reality? Dont
teachers have 'real world' skills? What world do they live in then?

I guess maths belongs to communities of practice, and there are many
different 'maths languages/genres', many different 'maths realities',
including maybe 'school maths', 'street maths' and maybe
'nursing maths', etc, etc. Im not sure if any one version
is any more 'real' than any other, though professional mathematicians
generally privelege their own version (and Ive often heard engineers
privelege theirs too.)

Some of these communities are more engaged in working ON mathematics
(is that what you mean by 'pure') and others in working 'WITH'
mathematics (is that what we mean by 'applied'?) But as we know,
breakdowns often occur which force us to switch our attention from
the 'with' to the 'on': (I usually cite Leont'ev for bringing this to
 attention, but I think I 'knew' of it before I read it n
Leont'ev... hah hah). In addition,almost all pure mathematical
activity can be considered to be 'applied' in the above sense:pure
maths uses itself recursively to build new
knowledge.

However, when a child plays at counting Im not sure I want to call
this 'pure' or 'applied'. And Im not convinced that the 'pure'
version entails 'understanding' while the 'applied' doesnt. Im
thinking of the CAD/CAM worker who uses the command G90 to set the
coordinates back to absolute reference coordinates: the worker may
not 'understand' the subroutine called, but they certainly know
and 'understand' coordinates, and how to use this to programme the
machine intelligently. The application requires a kind of 'external'
understanding of the maths and its relations with other objects,
perhaps without a complete internal understanding of the math, its
relations to its mathematical parts.

Generally the internal and
external understandings go together, and need each other, like the
child who playfully counts, also counts things playfully.

Surely the problem we need to address is the relative 'encapsulation'
of school maths.The child's first playful counting experiences can
too easily be replaced by and undermined by formal school learning
practices. We need to help the two to articulate: the 'street maths'
and the 'school maths' practices need to come to make joint sense.

I suppose most teachers speak some kind of 'school maths' and maybe
'street maths' and maybe some other dialects and genres. Maybe we can
say it would be helpful to pedagogy if we as teachers have a better
understanding of our pupils 'street maths' and 'school maths'.... and
the relations between the two which might help joint articulation

well? what do you think?

'> Date: Tue, 16 Oct 2001 09:25:26 +0100
> Subject: Re(2): ontologic math anxiety
> To: xmca@weber.ucsd.edu
> From: "Martin Owen" <mowen@rem.bangor.ac.uk>
> Reply-to: xmca@weber.ucsd.edu

> xmca@weber.ucsd.edu writes:
> >Shouldn't teachers be expected to actually
> >possess real world skills before they are sent into the classroom?
> But then why should we expect teachers to have had a better experience
> than anyone esle. The real world skill seems to be not very good at
maths.
> Maybe I should re-phrase:
> "the world is full of people who have not had a good experience of
> mathematics"
>
> Martin Owen
> Labordy Dysgu- Learning Lab
> Prifysgol Cymru Bangor- University of Wales, Bangor
>
> "How do you explain school to a higher intelligence?"
>
>



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